Quant Boosters - Rajesh Balasubramanian - Set 5



  • Let the roots be p, 1/p and q.
    p ∗ 1/p ∗ q = 2
    So, the 3rd root = q = 2
    p + 1/p + 2 = 4
    p + 1/p = 2 {For any real p |p + 1/p ≥ 2| }
    p^2 – 2p + 1 = 0
    (p – 1)^2 = 0. Hence, p is 1.
    The roots are 1, 1 and 2.
    (1 × 1) + (1 × 2) + (1 × 2) = 5 = m
    Answer choice (a)



  • Q21) x^4 – 4x^3 + ax^2 – bx + 1 = 0 has positive real roots. What is the maximum possible value of a + b?
    a) 20
    b) 12
    c) 8
    d) 10



  • Fabulous question.

    If we have an equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0 with roots p, q, r and s.
    We can say sum of the roots p + q + r + s = −b/a.
    Sum of the products taken two at a time pq + pr + ps + qr + qs + rs = c/a.
    Sum of the products taken three at a time pqr + pqs + prs + qrs = −d/a.
    Product of the roots pqrs = e/a.

    Note 1: We alternate between - and +
    Note 2: Start with the immediate lower power for sum of roots, stick a negative symbol and then alternate.

    So, if p, q, r, s were roots of this equation
    p + q + r + s = 4,
    pqrs = 1

    Or, Arithmetic mean of p,q, r, s = 1 and geometric mean of pqrs = 1.
    p, q, r s are positive real numbers. AM = GM. What does this mean?
    This means that all 4 numbers are equal.
    Or, this expression is (x-1)^4
    a = pq + pr + ps + qr + qs + rs = 6
    -(-b) = pqr + pqs + prs + qrs = 4
    a = 6, b = 4.
    a + b = 10



  • Q22) What is the minimum value of f(x) = x^2 – 5x + 41?
    a) 139/4
    b) 149/4
    c) 129/4
    d) 119/4



  • This question is present for one reason and one reason only. To talk about the idea of “completion of squares”. There are two ways of solving this question - The ugly differentiation based method and the beautiful completion of squares method. Always pick the elegant method. You might not prefer VVS Laxman over Gary Kirsten, or Federer over Nadal. But these are matters of sport. When it comes to math solutions – elegant solutions kick ass every time.

    What is this famous completion of squares method ? Any quadratic expression of the form x^2 + px + q can be written in the form (x +a)^2 + b.

    Write in that form, enjoy the equation and have some fun.

    x^2 – 5x + 41 = (x + a)^2 + b. what value should ‘a’ take? Forget about b for the time being.

    (x + a)^2 = x^2 + 2ax + a^2. The 2ax term should correspond to -5x. Done and dusted.

    a = (−5/2).
    a^2 = (25/4)

    x^2 – 5x + 41 can be written as x^2 – 5x + (25/4) – (25/4) + 41 = (x – (5/2))^2 + 41 – (25/4)

    = (x – (5/2))^2 + 139/4

    The minimum value this expression can take is 139/4.



  • Q23) Consider integers p, q, r such that |p| < |q| < |r| < 40. p + q + r = 20. What is the maximum possible value of pqr?
    a) 3600
    b) 3610
    c) 3510
    d) 3500



  • We need to maximise pqr. So,w e should shoot for either all three positive or two negative and one positive. Given p + q + r = 20, two positive and one negative works better. Let us say we choose p, q to be negative and r to be positive. Maximum value r can take is 39. p + q now becomes -19.

    p = -9, q = -10 and r = 39 works best.

    So, the product should be -9 * -10 * 39 = 3510.



  • Q24) Consider integers m, n such that -5 < m < 4 and -3 < n < 6. What is the maximum possible value of m^2 - mn + n^2?
    a) 65
    b) 60
    c) 50
    d) 61



  • Points to note

    1. m^2 and n^2 are always positive
    2. -mn will be maximum when m and n have opposite signs

    After this, it is pretty simple. We are better off with m being negative and n positive as in this case they can take higher values. m = -4, n = 5 works best. Note that as far as absolute value is concerned maximum |m| can be is 4 and maximum |n| can be is 5. This is important, otherwise we may have a scenario where even sacrificing –mn we might be able to maximise this.

    So, m = -4, n = 5 works
    m^2 -mn + n^2 = 16 + 20 + 25 = 61.



  • Q25) Consider three distinct positive integers a, b, c all less than 100. If |a - b| + |b - c| = |c – a|, what is the maximum value possible for b?
    a) 98
    b) 99
    c) 50
    d) 100



  • |q – p| is the distance between p and q on the number line. |p –q| is the same as |q –p| to begin with.

    So, in this case we are told |a -b| + |b -c| = |c – a|. Think about this. What does this mean? There are three points on the number line. We are talking about 3 distances on the number line here. We know that sum of some two of the distances is equal to the third. What does this tell us?

    This tells us that the point b has to be in between a and c. With this we are done. We can have a or c to be 99 and b to be 98.

    Maximum value b can take is 98. Classic question.



  • Q26) If a, b, c are integers such that – 50 < a, b, c < 50 and a + b + c = 30, what is the maximum possible value of abc?



  • abc will be maximum when it is positive. So, a, b, c can all be positive or two of the three can be negative and one positive.

    When all are positive, max product is when the numbers are 10, 10 and 10.

    When two are negative and one positive, the best–case scenario would be when two negative numbers are as low as possible (magnitudes as high as possible) so that the product can be high. Now, in order for the product to be maximum, the positive number should be as high as possible. So, let the positive number be 49. Then the sum of the two negative numbers should be –19. The best– case scenario would be when numbers are 49, –9, –10.

    Product would be 4410. Answer choice (c)



  • Q27) a, b, c are distinct natural numbers less than 25. What is the maximum possible value of |a – b| + |b – c| – |c – a|?



  • 0_1516704238049_816f40c0-8153-40ed-a6df-affb92344029-image.png

    For any two points M, N on the number line representing numbers m, n the distance MN = | m - n|.
    So, for three points, P, Q and R on the number line |p – q|, |q – r|, |r – p| are distances between three pairs of points on the number line.

    In this case, we are trying to find maximum value of |a – b| + |b – c| – |c – a|. If b lies between a and c, the above value would be zero. So, b should not be between a and c.
    The best case scenario would be if a, c were very close to each other and far from b. Let us try b = 24, a = 1, c = 2.
    In this case |a – b| + |b – c| – |c – a| = 23 + 22 – 1 = 44. This is the maximum possible value.

    We could also have b = 1, a = 24, c = 23,
    |a – b| + |b – c| – |c – a| = 23 + 22 – 1 = 44.
    Answer choice (a)



  • Q28) Consider integers p, q such that – 3 < p < 4, – 8 < q < 7, what is the maximum possible value of p^2 + pq + q^2?



  • Trial and error is the best approach for this question. We just need to be scientific about this.
    p^2 and q^2 are both positive and depend on |p| and |q|. If p, q are large negative or large positive numbers, p^2 and q^2 will be high.
    pq will be positive if p, q have the same sign, and negative if they have opposite signs.

    So, for p^2 + pq + q^2 to be maximum, best scenarios would be if both p & q are positive or both are negative.

    Let us try two possibilities.
    p = – 2, q = – 7: p^2 + pq + q^2 = 4 + 14 + 49 = 67
    p = 3, q = 6: p^2 + pq + q^2 = 9 + 15 + 36 = 60
    Whenever we have an expression with multiple terms, there are two key points to note.

    The equation will be most sensitive to the highest power.
    The equation will be more sensitive to the term with the greater value.

    In the case, q.

    In this question, we have a trade–off between higher value for p^2 and q^2. For q^2, the choice is between 6^2 and (–7)^2. This impact will overshadow the choice for p (where we are choosing between –2 and 3).

    So, the maximum value for the expression would be 67.
    Answer choice (B)



  • Q29) If a, b, c are distinct positive integers, what is the highest value a * b * c can take if a + b + c = 31?



  • The sum of three numbers is given; the product will be maximum if the numbers are equal.

    (a + b + c)/3 > (abc)^(1/3)

    So, if a + b + c is defined, abc will be maximum when all three terms are equal. In this instance, however, with a, b, c being distinct integers, they cannot all be equal.

    So, we need to look at a, b, c to be as close to each other as possible.
    a = 10, b =10, c = 11 is one possibility, but a, b, c have to be distinct. So, this can be ruled out.
    The close options are,
    a, b, c : 9, 10, 12; product = 1080
    a, b, c : 8, 11, 12: product = 1056
    Maximum product = 1080



  • Q30) a and b are roots of the equation x^2 - px + 12 = 0. If the difference between the roots is at least 12, what is the range of values p can take?


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